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Statistics of the Voronoi cell perimeter in large
bi-pointed maps
Emmanuel Guitter
To cite this version:
Emmanuel Guitter. Statistics of the Voronoi cell perimeter in large bi-pointed maps. Journal of
Statistical Mechanics: Theory and Experiment, IOP Publishing, 2018, 18, pp.073409.
�10.1088/1742-5468/aacfbc�. �hal-01646351�
BI-POINTED MAPS
EMMANUEL GUITTER
Abstract. We study the statistics of the Vorono¨ı cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance 2s and their Vorono¨ı cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of s for maps with a large given volume N both in the scaling limit where s scales as N1/4, in which case the Vorono¨ı cell perimeter scales as N1/2, and
in the local limit where s remains finite, in which case the perimeter scales as s2for large
s. The obtained laws are universal and are characteristics of the Brownian map and the Brownian plane respectively.
1. Introduction
The statistics of planar maps, which are graphs embedded in the sphere considered up to continuous deformations, has raised much attention over the years and its study gave rise to a myriad of combinatorial results which describe these mathematical objects at the discrete level, as well as of probabilistic results which characterize their universal continuous limits such as, depending on the underlying scaling regime, the Brownian map or the Brownian plane. Among refined descriptions of planar maps is the characterization of their Vorono¨ı cellswhich, for maps with a number of distinguished marked vertices, are the domains which, using the natural graph distance, regroup vertices according to which marked vertex they are closer to. The simplest realization of Vorono¨ı cells is for bi-pointed maps, i.e. maps with exactly two marked distinct and distinguished vertices, say v1 and v2, in which case the
map is naturally split into two connected complementary domains gathering vertices closer to v1 than to v2 and conversely. A natural question is then to characterize the statistics of
these two Vorono¨ı cells in an ensemble where, say the total volume (equal for instance to the number N of faces) of the map is fixed. A first result was obtained in [6] where the law for the proportion φ of the total volume N (0≤ φ ≤ 1) spanned by one of the cells was obtained exactly in the limit of large N . It was found that this law is uniform between 0 and 1, proving a particular instance of some more general conjecture by Chapuy for Vorono¨ı cells within maps of fixed genus [5]. Another result concerned the characterization of Vorono¨ı cells for bi-pointed maps where the graph distance d(v1, v2) is also fixed [7]. In the limit
where N→ ∞ and d(v1, v2) is kept finite, it was shown that exactly one of the two cells has
an infinite volume, while the other remains finite. The universal law for this finite Vorono¨ı cell volume in the limit of large d(v1, v2) was derived in [7].
In this paper, we address another question about bi-pointed maps by looking at their Vorono¨ı cell perimeter, which is the length L of the frontier which separates the two Vorono¨ı cells. We again consider the ensemble of bi-pointed maps with a fixed volume N and a fixed distance d(v1, v2). Explicit universal expressions are obtained for the Vorono¨ı cell perimeter
statistics in the following two regimes for which N→ ∞: (1) the so-called scaling limit where d(v1, v2) scales as N1/4, and (2) the so called local limit where d(v1, v2) remains finite. In case
(1), whose behavior characterizes the so-called Brownian map, the Vorono¨ı cell perimeter scales as N1/2 and an explicit universal law is derived for the properly rescaled perimeter
L/N1/2. In case (2), whose behavior for large d(v
1, v2) characterizes the so-called Brownian
plane, a universal non-trivial law is obtained if we now rescale the perimeter L by d(v1, v2)2. 1
The paper is organized as follows: we first discuss in Section 2 the maps under study and give a precise definition of their Vorono¨ı cell perimeter, whose statistics is encoded in some appropriate generating function (Section 2.1). To this generating function is naturally associated some scaling function, introduced in Section 2.2, whose determination is the key of our study as it may be used to encode the N → ∞ limit of the Vorono¨ı cell perimeter statistics. Section 3 is devoted to the computation of this scaling function thanks to a series of differential equations (Section 3.1) solved successively in Sections 3.2 and 3.3. The net result is an explicit expression for the desired scaling function which is then used in Sections 4 and 5 to address the Vorono¨ı cell perimeter statistics in detail. Section 4 deals with the scaling limit (1) introduced above. Section 4.1 explains in all generality how to obtain expected values characterizing the Vorono¨ı cell perimeter in this regime directly from the scaling function while Section 4.2 presents the corresponding explicit formulas and plots. Section 5 deals with the local limit (2): here again, we first explain in Section 5.1 the general formalism which connects the local limit statistics to the already computed scaling function. This is then used in Section 5.2 to provide a number of laws, which we make explicit and/or plot for illustration. We present our conclusions in Section 6 while some technical intermediate computations are discussed in Appendix A.
2. Bi-pointed maps and Vorono¨ı cell perimeter
2.1. Generating functions. This paper deals with the statistics of bi-pointed planar quad-rangulations, which are planar maps whose all faces have degree 4, with two marked distinct vertices, distinguished as v1 and v2, at some even graph distance from each other. Such
maps are naturally split into two Vorono¨ı cells, which are complementary connected do-mains gathering vertices which are closer to one marked vertex than to the other. A precise definition of the Vorono¨ı cells is given by the Miermont bijection [8] which provides a one-to-one correspondence between the above defined bi-pointed planar quadrangulations and so-called planar iso-labelled two-face maps (i-l.2.f.m). Those are planar maps with exactly two faces, distinguished as f1 and f2, and whose vertices carry positive integer labels. The
labels are required to satisfy the following three constraints: (L1) labels on adjacent vertices differ by 0 or±1;
(L2) the minimum label for the set of vertices incident to f1 is 1;
(L3) the minimum label for the set of vertices incident to f2 is 1.
As it appears in the Miermont bijection, the vertices of the i-l.2.f.m are in one-to-one corre-spondence with the unmarked (i.e. other than v1and v2) vertices in the associated bi-pointed
quadrangulation and their label corresponds precisely to their graph distance in the quad-rangulation to the closest marked vertex v1 or v2. Moreover, vertices incident to f1 (resp.
f2) in the i-l.2.f.m correspond to vertices closer to v1 than to v2 (resp. closer to v2 than to
v1) in the quadrangulation. As a consequence, we define the Vorono¨ı cells of the bi-pointed
quadrangulation as the two domains spanned by f1 and f2 respectively upon drawing the
i-l.2.f.m and the associated quadrangulation on top of each other (see figure 1). Each face of the quadrangulation is actually traversed by exactly one edge of the i-l.2.f.m. These edges form a simple closed loopL separating the two faces f1 and f2, together with a number of
subtrees attached to the vertices ofL, on both sides of the loop. Subtree edges of either side correspond to faces in the bi-pointed quadrangulation which lie within a single Vorono¨ı cell, while loop edges correspond instead to faces covered by both cells. The length L (=number of edges) of the loopL is called the Vorono¨ı cell perimeter as it is a natural measure of the size of the boundary between the two cells.
In this paper, we shall study the statistics of L for maps with a fixed value d(v1, v2) = 2s
(s≥ 1) of the graph distance between the two marked vertices v1 andv2. In the associated
i-l.2.f.m language, this distance is fixed via the following fourth requirement: (L4) the minimum label for the set of vertices incident toL is s.
1
0
2
1
2
0
1
2
1
2
2
1
3
1
1
2
2
2
v
1
v
2
f
1
f
2
L
Figure 1. An example of planar bi-pointed quadrangulation (thin black edges) whose two marked vertices v1 and v2 are at even distance 2s (here
s = 2). Each vertex is labelled by its distance to the closest marked vertex v1
or v2. Applying the Miermont bijection transforms the quadrangulation into
an i-l.2.f.m (thick blue edges) whose two faces f1and f2define two Vorono¨ı
cells (here the Vorono¨ı cell corresponding to f2 has been filled with light
blue). The Vorono¨ı cell perimeter L is the length of the loopL separating the two faces of the i-l.2.f.m (i.e. the boundary length of the light blue domain, here L = 3).
w
1w
2 0 0labels
≥ 1−s
labels
≥ 1−t
g
labels
≥ 0
g z
g
left part:
right part:
spine:
Figure 2. A schematic picture of a l.c. enumerated by Xs,t(g, z), see text.
For a fixed graph distance d(v1, v2) = 2s, we may keep a control on both the volume
N (= number of faces) of the planar bi-pointed quadrangulations and on their Vorono¨ı cell perimeter L by considering their generating function Fs(g, z) with a weight gNzL. Via the
Miermont bijection, Fs(g, z) is also the generating function of i-l.2.f.m satisfying (L4) with
a weight gEzL, where E is the number of edges of the i-l.2.f.m and L the length of the loop
L.
As explained in [6] in a slightly different context, the generating function Fs(g, z) is
closely related to some other generating function, Xs,t(g, z), of properly weighted labelled
chains (l.c), which are planar one-face maps whose vertices carry integer labels and with two distinct (and distinguished) marked vertices w1 and w2. The unique shortest path in
spine, the map consists of (possibly empty) subtrees attached to the left and to the right of each internal spine vertex and at the marked vertices w1 and w2 (see figure 2). Given two
positive integers s and t, Xs,t≡ Xs,t(g, z) is the generating function of planar l.c subject to
the following constraints:
(`1) labels on adjacent vertices differ by 0 or ±1;
(`2) w1 and w2 have label 0. The minimum label for spine vertices is 0. Spine edges receive
a weight g z;
(`3) the minimum label for vertices of the left part, formed by the subtree attached to w1
and all the subtrees attached to the left of any internal spine vertex (with the spine oriented from w1to w2), is larger than or equal to 1− s. The edges of all these subtrees
receive a weight g;
(`4) the minimum label for vertices of the right part, formed by the subtree attached to w2
and all the subtrees attached to the right of any internal spine vertex, is larger than or equal to 1− t. The edges of all these subtrees receive a weight g;
For convenience, Xs,tincludes a first additional term 1 and, for s, t > 0, we set Xs,0 = X0,t=
X0,0= 1.
By some appropriate decomposition of the i-l.2.f.m, it is then easy to relate Fs(g, z) to
Xs,t(g, z) (see [6] for a detailed description of the decomposition) via:
(1) Fs(g, z) = ∆s∆tlog(Xs,t(g, z)) t=s= log X s,s(g, z)Xs−1,s−1(g, z) Xs−1,s(g, z)Xs,s−1(g, z) ,
where ∆s denotes the finite difference operator (∆sf (s)≡ f(s) − f(s − 1)). In brief, the
above relation is obtained by first shifting all the labels of the i-l.2.f.m by−s so that the minimum label on the loop L becomes 0 and the minimum label within f1 becomes 1− s,
as well as the minimum label within f2. One then releases the constraints by demanding
that the minimum label within f1 be≥ 1 − s and that within f2 be≥ 1 − t for some t not
necessarily equal to s. The loop of the obtained two-face map forms (by cutting it at each loop vertex labelled 0) a cyclic sequence of particular l.c with no internal spine vertex labelled 0. Since linear (i.e. non-cyclic) sequences of such l.c are precisely enumerated by Xs,t(g, z),
the desired cyclic sequences are easily seen to be enumerated by log(Xs,t(g, z)). Applying
the finite difference operator ∆srestores the constraint that the minimum label within f1is
exactly 1− s, and similarly, the finite difference operator ∆trestores the constraint that the
minimum label within f2is exactly 1− t, which eventually equals the desired value 1 − s by
setting t = s.
It is therefore sufficient to obtain an expression for Xs,t(g, z) to determine the desired
Fs(g, z). The generating function Xs,t(g, z) itself is fully determined by its relation with
the generating function Rs(g) for so called well-abelled trees1(accounting for the subtrees
attached to the spine), namely (see [6]):
(2) Xs,t(g, z) = 1 + g z Rs(g)Rt(g)Xs,t(g, z) (1 + g z Rs+1(g)Rt+1(g)Xs+1,t+1(g, z))
for s, t≥ 0. As for the generating function Rs(g) itself (which depends on g only), its explicit
expression is well-known and takes the parametric form [4]:
(3) Rs(g) = 1 + 4x + x2 1 + x + x2 (1− xs)(1 − xs+3) (1− xs+1)(1− xs+2) for g = x 1 + x + x2 (1 + 4x + x2)2 .
Here x varies in the range 0≤ x ≤ 1 so that g varies in the range 0 ≤ g ≤ 1/12 (ensuring a proper convergence of the generating function at hand). From the explicit form (3) and the equations (2) and (1), we may in principle determine Xs,t(g, z) and Fs(g, z). For instance,
1To be precise, R
s(g) enumerates planted trees with vertices labeled by integers satisfying (`1), with root
for z = 1, the solution of (2) reads (4) Xs,t(g, 1) = (1− x3)(1 − xs+1)(1 − xt+1)(1 − xs+t+3) (1− x)(1 − xs+3)(1− xt+3)(1− xs+t+1) so that (5) Fs(g, 1) = log ( 1− x2s+3 1 − x2s2 (1− x2s−1) (1− x2s+2)2 ) .
For arbitrary z however, we have not been able to obtain such explicit expressions for Xs,t(g, z) or Fs(g, z) and we therefore have no prediction on the statistics of Vorono¨ı cell
perimeters for a finite volume N of the quadrangulation. Fortunately, as explained in de-tails in the next section, all the above generating functions have appropriate scaling limits whose knowledge will allow us to characterize the statistics of Vorono¨ı cell perimeters in quadrangulations with a large or infinite volume N .
Before turning to scaling functions, let us end this section by looking, now for arbitrary z, at the limit
X∞,∞(g, z)≡ lim
s,t→∞Xs,t(g, z) .
It is easily obtained from (2) as the solution of X∞,∞(g, z) = 1 + z x 1 + x + x2X∞,∞(g, z) 1 + z x 1 + x + x2X∞,∞(g, z) , which tends to 1 when g→ 0, namely
(6) X∞,∞(g, z) =
(1+x+x2)1+x+x2
−z x−p(1+x+x2−z x)2− 4 x2z2
2 x2z2
with the same parametrization of g as in (3). Note that X∞,∞(g, z) is well defined for
0≤ z ≤ (1 + x + x2)/(3x), hence for 0
≤ z ≤ 1 in the whole range 0 ≤ g ≤ 1/12 (0 ≤ x ≤ 1). 2.2. Scaling functions. The generating functions Rs(g), Xs,t(g, z) and Fs(g, z) are
singu-lar when g and z tend simultaneously toward their critical value 1/12 and 1 respectively. The corresponding singularities are encoded in scaling functions, as defined below, whose knowledge will allow us to describe the statistics of Vorono¨ı cell perimeters in quadrangula-tions whose volume N is large. A non trivial scaling regime is obtained by letting g and z approach their critical values as
(7) g = G(a, )≡ 1 12 1−a 4 36 4 , z = Z(λ, )≡ 1 − λ 2
with → 0, and letting simultaneously s and t become large as s =S
, t = T
,
with a, λ, S and T finite positive reals. From (3), we have for instance the expansion (8) RbS/c(G(a, )) = 2 + r(S, a) 2+ O(3) , r(S, a) =−
a2 1 + 10e−aS+ e−2aS
3 (1− e−aS)2 .
Expanding (2) at second order in , we have similarly
XbS/c,bT /c(G(a, ), Z(λ, )) = 3 + x(S, T, a, λ) + O(2) ,
where the scaling function x(S, T, a, λ) is solution of the non-linear partial differential equa-tion
(9) 2 x(S, T, a, λ)2+ 6 ∂Sx(S, T, a, λ) + ∂Tx(S, T, a, λ)
(with the short-hand notations ∂S ≡ ∂/∂S and ∂T ≡ ∂/∂T ). For instance, from the explicitly
expression (4) at z = 1 (i.e. λ = 0), we immediately deduce
x(S, T, a, 0) =−3 a − 6a e
−aS+ e−aT
− 3e−a(S+T )+ e−2a(S+T )
(1− e−aS) (1− e−aT) 1− e−a(S+T ) ,
which, as easily checked, satisfies (9) when λ = 0. Finally, eq. (1) implies that
(10) FbS/c(G(a, ), Z(λ, )) =F(S, a, λ) 3+ O(4) where F(S, a, λ) = 13 ∂S∂Tx(S, T, a, λ) T =S .
The next section is devoted to the actual computation of the scaling functionF(S, a, λ). We will then show in the following sections how to use this result to address the question of the Vorono¨ı cell perimeter statistics in large maps.
3. Computation of the scaling function F(S, a, λ)
3.1. Differential equations. We have not been able to solve the equation (9), as such, when λ6= 0 and therefore have no explicit expression for x(S, T, a, λ) for arbitrary λ and for independent S and T . Such a formula is however not necessary to computeF(S, a, λ) as we may instead recourse to a sequence of two ordinary differential equations, both inherited from (9), which determine successively the following two quantities:
(11) Y (S, a, λ)≡ x(S, T, a, λ)|T =S , H(S, a, λ)≡ ∂S∂Tx(S, T, a, λ)|T =S .
Note that these quantities depend on a single “distance parameter” S instead of two (since we always eventually fix T = S). Note also that the knowledge of H(S, a, λ) is exactly what we look for since, from (10), we have
F(S, a, λ) =13H(S, a, λ) .
Since d
dSY (S, a, λ) = ∂Sx(S, T, a, λ) + ∂Tx(S, T, a, λ)
|T =S, we immediately deduce from
(9) the following differential equation for Y (S, a, λ):
(12) 2 Y (S, a, λ)2+ 6 d
dSY (S, a, λ) + 54r(S, a) = 54λ .
This equation, with appropriate boundary conditions, will allow us to compute Y (S, a, λ) explicitly.
Similarly, differentiating (9) with respect to S and T and then setting T = S, we deduce
(13) 4Y (S, a, λ) H(S, a, λ) + d dSY (S, a, λ) 2 + 6 d dSH(S, a, λ) = 0 ,
where we used the identity ∂Sx(S, T, a, λ)|T =S = ∂Tx(S, T, a, λ)|T =S = 12dSd Y (S, a, λ),
ob-vious by symmetry2. Knowing Y (S, a, λ), this equation will determine H(S, a, λ) by some
appropriate choice of boundary conditions.
Let us now solve the two equations (12) and (13) successively.
2We indeed clearly have X
3.2. Solution of (12). The non-linear differential equation (12) is of the Riccati type and is therefore easily transformed, by setting
(14) Y (S, a, λ) = 3 d
dSlog V (S, a, λ)
,
into a second order homogeneous linear differential equation d2
dS2V (S, a, λ) + 3(r(S, a)− λ)V (S, a, λ) = 0 .
The solution of this equation is easily found to be (15) V (S, a, λ) = eV (σ, A) with σ≡ e−aS and A≡
r 1 +3λ a2 where eV (σ, A) is solution of σ d dσV (σ, A) + σe 2 d2 dσ2V (σ, A) =e A2+ 12σ (1− σ)2 e V (σ, A) , hence given by e V (σ, A) = V0 (1− σ)3 σ AW (σ, A) − α σ−AW (σ,−A) , W (σ, A) = 4A3(1 − σ)3+ 12A2(1 − σ)2(1 + σ) + A(1 − σ)(11 + 38σ + 11σ2) + 3(1 + σ)(1 + 8σ + σ2) .
Note that, apart from the global arbitrary multiplicative constant V0 which is unimportant
as it drops out in (14), this solution involves some unknown constant α which needs to be fixed. To this end, we recall that, for s and t tending to∞, Xs,t(g, z) tends to X∞,∞(g, z)
given by (6). In particular, we have the expansion X∞,∞(G(a, ), Z(λ, )) = 3− 3
p
a2+ 3λ + O(2)
from which we deduce
Y (S, a, λ)S→∞∼ −3pa2+ 3λ =−3a A
for a > 0. Sending S→ ∞ for a > 0 amounts to letting σ → 0. From (14), we must therefore ensure −3a e σ V (σ, A) d dσV (σ, A)e σ→0→ −3a A ⇐⇒ σ e V (σ, A) d dσV (σ, A)e σ→0→ A .
Since W (σ, A) has a finite limit when σ → 0, and since A > 0, this latter requirement is fulfilled only for α = 0 (in all the other cases, the limit is−A instead). This fixes the value of the parameter α to be 0, while we may set V0 = 1 without loss of generality, so that we
eventually get the expression
(16) e V (σ, A) = σ AW (σ, A) (1− σ)3 , W (σ, A) = 4A3(1 − σ)3+ 12A2(1 − σ)2(1 + σ) + A(1 − σ)(11 + 38σ + 11σ2) + 3(1 + σ)(1 + 8σ + σ2) and, from (14), (17)
Y (S, a, λ) = a eY (σ, A) with σ≡ e−aS and A
≡ r 1 + 3λ a2 , e Y (σ, A) =−3 σ e V (σ, A) d dσV (σ, A) =e −3 A − 36 σ U (σ, A) (1− σ) W (σ, A) , U (σ, A) = 2A2(1 − σ)2+ 5A(1 − σ)(1 + σ) + 3(1 + 3σ + σ2)
and W (A, σ) as in (16).
3.3. Solution of (13). Given Y (S, a, λ), the solution of the linear first order equation (13) is given by the general formula
H(S, a, λ) = e−23 RS S0dT Y (T ,a,λ) ( H(S0, a, λ)− 1 6 Z S S0 dT d dTY (T, a, λ) 2 e23 RT S0dT 0Y (T0,a,λ) )
for some arbitrary S0> 0. Here the value of H(S0, a, λ) is some integration constant which
may a priori be chosen arbitrarily. As we shall see below, it will be fixed in our case by the requirement that H(S, a, λ) has the desired large S behavior. It is straightforward from (14) that e−23 RS S0dT Y (T ,a,λ)= V (S 0, a, λ) V (S, a, λ) 2
with V (S, a, λ) as in eqs. (15) and (16). We may therefore write the solution of (13) as
H(S, a, λ) = 1 (V (S, a, λ))2 (V (S0, a, λ)) 2 H(S0, a, λ)− 1 6 Z S S0 dT V (T, a, λ) d dTY (T, a, λ) !2 which, upon changing again variables by setting
H(S, a, λ) = a3H(σ, A) with σe
≡ e−aS and A≡ r
1 +3λ a2
and upon using eqs. (16) and (17), rewrites explicitly e H(σ, A) = (1− σ) 6 σ2A(W (σ, A))2 ( σ2A 0 (W (σ0, A))2 (1− σ0)6 e H(σ0, A) + Z σ σ0 dτ M (τ, A) ) , M (σ, A) = 216 σ 2A+1(P (σ, A))2 (1− σ)10(W (σ, A))2 , P (σ, A) =W (σ, A)(1− σ)2 d dσ σ U (σ, A) (1− σ) W (σ, A) = 8A5(1 − σ)5(1 + σ) + 4A4(1 − σ)4(11 + 20σ + 11σ2) + 2A3(1 − σ)3(1 + σ)(47 + 86σ + 47σ2) + A2(1 − σ)2(97 + 386σ + 594σ2+ 386σ3+ 97σ4) + 6A(1− σ)(1 + σ)(8 + 33σ + 68σ2+ 33σ3+ 8σ4) + 9 (1 + 6σ + 27σ2+ 32σ3+ 27σ4+ 6σ5+ σ6) ,
where σ0= e−aS0. Remarkably enough, the function M (σ, A) above has an explicit primitive
(in σ) given by:
C(σ, A) = 12 35σ 2A+2 ( Q(σ, A) (1− σ)9W (σ, A) − A(A − 1)(A2 − 1)(4A2 − 1)(4A2 − 9) 2F1(1, 2A + 2, 2A + 3, σ) )
where2F1(1, 2A + 2, 2A + 3, σ) is a simple hypergeometric function
(18) 2F1(1, 2A + 2, 2A + 3, σ) = (2A + 2)
X
n≥0
σn
and where Q(σ, A) a polynomial of degree 11 in σ given by
(19) Q(σ, A) =
11
X
i=0
qi(σ)A11−i(1− σ)11−i
with q0(σ) = 64 , q1(σ) = 32(4 + 7σ) , q2(σ) = 80 (−3 + 13σ) q3(σ) = 48(−12 + 3σ + 39σ2− 17σ3) , q4(σ) = 12(17− 278σ + 400σ2− 90σ3− 65σ4) q5(σ) = 6(980 + 1365σ + 424σ2+ 1374σ3− 876σ4+ 109σ5) q6(σ) = (35363 + 88857σ + 75432σ2+ 45698σ3− 1965σ4− 2715σ5+ 1250σ6) q7(σ) = (100336 + 279172σ + 377163σ2+ 267688σ3+ 118786σ4− 12360σ5+ 2611σ6+ 244σ7) q8(σ) = 2(74901 + 220667σ + 395067σ2+ 388137σ3+ 222947σ4+ 77061σ5− 2895σ6 + 1355σ7 − 240σ8) q9(σ) = 3(40659 + 122280σ + 287397σ2+ 350940σ3+ 284873σ4+ 126322σ5+ 37701σ6 + 724σ7+ 206σ8 − 102σ9) q10(σ) = 9(5673 + 16595σ + 51926σ2+ 83180σ3+ 83930σ4+ 50678σ5+ 17870σ6+ 4994σ7 + 155σ8+ 5σ9 − 6σ10) q11(σ) = 27(315 + 840σ + 3465σ2+ 9576σ3+ 6510σ4+ 9864σ5+ 3150σ6+ 1000σ7 + 279σ8+ σ10) .
Knowing C(σ, A), we may thus write e H(σ, A) = (1− σ) 6 σ2A(W (σ, A))2 ( σ2A 0 (W (σ0, A)) 2 (1− σ0)6 e H(σ0, A) + C(σ, A)− C(σ0, A) ) ,
leaving us with the determination of the free integration constant eH(σ0, A) = H(S0, a, λ)/a3.
From its combinatorial interpretation, Fs(g, z) is expected to vanish at large s (see for
instance (5) when z = 1) and so is thus F(S, a, λ) = (1/3)H(S, a, λ) at large S. Sending S→ ∞ amounts (for a > 0) to let σ → 0, in which case we find
C(σ, A) = σ2A108 (A + 1)(2A + 3)2σ2+ O(σ3).
Demanding that H(S, a, λ)→ 0 at large S forces us to choose the (so far arbitrary) value e
H(σ0, A) such that σ2A
0 (W (σ0,A))2
(1−σ0)6 H(σe 0, A)− C(σ0, A) = 0 in order to avoid a divergence of e
H(σ, A) as σ−2A when σ→ 0. This leads to the final expression3
(20) e H(σ, A) = (1− σ) 6C(σ, A) σ2A(W (σ, A))2 =12 35σ 2 ( Q(σ, A) (1− σ)3(W (σ, A))3 − A(A − 1)(A2 − 1)(4A2 − 1)(4A2 − 9) (1− σ) 6 (W (σ, A))2 2F1(1, 2A + 2, 2A + 3, σ) )
with W (A, σ) as in (16) and Q(σ, A) as in (19) above. The desired scaling functionF(S, a, λ) is then obtained directly via
(21) F(S, a, λ) =13a3H(σ, A) ,e σ≡ e−aS , A
≡ r
1 +3λ a2 . 3At small σ, we then have the expansion eH(σ, A) = 108 σ2
a
4
0
−∞
−∞
a
0
C
g
1 120
Figure 3. The contour integral over g used to extract the coefficient gN of, say Fs(g, z). In the g plane, the contour around 0 is deformed so as to
surround the cut of Fs(g, z) for real g > 1/12. By setting g = G(a, N−1/4),
the excursion around the cut corresponds in the a4variable to a path around
the negative real axis which makes a back and forth excursion from and to −∞ in the limit N → ∞. In the a plane, the corresponding path C is made of two straight lines at 45◦ from the real axis, oriented as shown from and
to infinity.
4. Large maps in the scaling limit
4.1. Quadrangulations with a large fixed volume. Fixing the value N of the volume of the quadrangulations amounts to extracting the coefficient gN of the various generating
functions. For instance, the number of bi-pointed planar quadrangulations of fixed volume N and with a fixed distance 2s between the marked vertices is directly given by
[gN]F s(g, 1) .
As for the statistics of the Vorono¨ı cell perimeter L within bi-pointed quadrangulations of fixed volume N and fixed distance 2s between the marked vertices, it is fully characterized by considering the expected value EN,s(zL) of zL in this ensemble, for 0 ≤ z ≤ 1. This
expectation is directly given by the simple ratio (22) EN,s(zL) = [gN]F s(g, z) [gN]F s(g, 1) .
In practice, extracting the coefficient gN may be performed via a contour integral over g
around 0, which may then be transformed into an integral over the variable a such that g = Ga, N−1/4= 1 12 1− a 4 36 N .
The integral over a runs on some contourCN, which, at large N , eventually tends to some
simple contourC made of two straight lines at 45◦ from the real axis, as displayed in figure
3.
The scaling limit corresponds to let N tend to infinity while looking at distances 2s of order N1/4. This is achieved by setting s = S N1/4 where S is kept finite, in which case we
have for instance, for z = 1: [gN ]FbS N1/4c(g, 1) = 1 2iπ I dg gN +1FbS N1/4c(g, 1) = 1 2iπ 12N N Z CN −da a3 9 1 1− a4 36 N N +1FbS N1/4c G(a, N−1/4), 1 ∼ N →∞ 1 2iπ 12N N7/4 Z C −da a3 9 e a4 36F (S, a, 0)
where we have used the expansion (10) (at z = 1, i.e. at λ = 0) with = 1/N1/4. Dividing4
the quantity FbS N1/4c(g, 1) × N
1/4dS by the large N asymptotic number512N/(4√πN3/2)
of bi-pointed quadrangulations having their two marked vertices at (arbitrary) even distance from each other leads to the probability ρ(S)dS that two marked vertices chosen uniformly at random at some even distance from each other in a quadrangulation of large volume N have their half-distance between S N1/4 and (S + dS) N1/4. The quantity ρ(S) is nothing
but the so-called distance density profile of the Brownian map, obtained here in the context of bi-pointed quadrangulations, and is thus given by
(23) ρ(S) = 2 i√π Z C −da a3 9 e a4 36F (S, a, 0) = 2 i√π Z ∞ 0 2t dt e−t2nFS, (1− i)√3t, 0− FS, (1 + i)√3t, 0o
upon changing variable and parametrizing the contourC by setting a = (1 ∓ i)√3t, with t real and positive.
We may compute in a similar way the quantity [gN]FbS N1/4c g, Z(λ, N−1/4)= 1 2iπ I dg gN +1FbS N1/4c g, Z(λ, N−1/4) = 1 2iπ 12N N Z CN −da a3 9 1 1− a4 36 N N +1FbS N1/4c G(a, N−1/4), Z(λ, N−1/4) ∼ N →∞ 1 2iπ 12N N7/4 Z C −da a3 9 e a4 36F (S, a, λ) . Using Z(λ, N−1/4)L ∼ e−λ L/N1/2
at large N , we deduce, after normalization, that
(24) EN,bS N1/4c e−λ L/N1/2 ∼ N →∞ ρ(S, λ) ρ(S) , ρ(S, λ) = 2 i√π Z ∞ 0 2t dt e−t2n FS, (1− i)√3t, λ− FS, (1 + i)√3t, λo
and ρ(S) = ρ(S, 0) as in (23) above. In the scaling limit, a non-trivial law is therefore obtained for the rescaled perimeter L/N1/2. Let us now examine this law in more details.
4.2. Explicit expressions and plots. Inserting the explicit expression (21) ofF(S, a, λ) into (24) allows to write ρ(S, λ) as a simple integral over t which may be evaluated nu-merically. Note that for S and λ real positive, ρ(S, λ) is actually real. Dividing ρ(S, λ) by ρ(S) = ρ(S, 0) allows us, from (24), to then evaluate the large N limit of the quantity EN,bS N1/4c
e−λ L/N1/2
as a function of S and λ. This quantity is plotted for illustration in figure 4 and, by definition, lies between 0 and 1. Let us now characterize it in more details via a number of more explicit expressions. ExpandingF(S, a, λ) at first order in λ, we get
F(S, a, λ) = F(S, a, 0) + λ × ∂λ∂ F(S, a, 0) + O(λ2 ) , F(S, a, 0) = 2a 3σ2 1 + σ2 (1− σ2)3 with σ = e −a S , ∂ ∂λF(S, a, 0) = − aσ2 385 − 189σ + 154σ2+ 54σ3 − 11σ4 − 9σ5 70(1− σ)(1 + σ)4 . 4
In FbS N1/4c(g, 1), the half-distance is fixed at some precise integer value close to S N1/4. Letting instead
the half-distance vary between S N1/4and (S + dS) N1/4
requires considering the quantity FbS N1/4c(g, 1) ×
N1/4dS.
lim
N→∞
E
N,b
S N1/4c
e
−λ L/N1/2!
λ
S
Figure 4. A plot of the large N limit of EN,bS N1/4c
e−λ L/N1/2 as a function of S and λ, as given by (24).
ρ(S)
µ(S)
S
S
Figure 5. Plots of ρ(S) and µ(S), as given by (25) and (27) respectively.
From the explicit expression ofF(S, a, 0), we may write via (23) a rather explicit expression for ρ(S) in the form of an integral over t. Introducing the short hand notations
ch = cosh(√3t S) , sh = sinh(√3t S) , co = cos(√3t S) , si = sin(√3t S) , we find (25) ρ(S) = r 3 π Z ∞ 0 dt 24t5/2e−t2 1 (co2− ch2)3× n ch sh 2co2 −1 co2+ch2 −1−1+co si 2ch2 −1 co2+ch2 −1−1 o. This function is plotted in figure 5 and the above formula matches already known expressions in the literature [4] for the distance density profile of the Brownian map. We have in particular the small S behavior
ρ(S) = 48 7 S
3
S
lim
N→∞
E
N,b
S N1/4c
L/N
1/2Figure 6. A plot of the large N limit of EN,bS N1/4c L/N
1/2, as given
by (25)–(26)–(27). We also indicated its limiting small S linear behavior, as given by (28), and its large S power law behavior, as given by (31).
More interesting is the linear term in λ: expanding eq. (24) at first order in λ yields indeed
(26) EN,bS N1/4c L/N1/2 ∼ N →∞ µ(S) ρ(S) µ(S) = 2 i√π Z ∞ 0 2t dt e−t2 −∂λ∂ FS, (1− i)√3t, 0+ ∂ ∂λF S, (1 + i)√3t, 0 .
From the above expression for ∂
∂λF (S, a, 0), we arrive at the following expression (with the
same short hand notations as before):
(27) µ(S) = r 3 π Z ∞ 0 dt 4 35t 3/2 e−t2 (
16 (co−si)(ch−sh) − 9 (co2
−2co si−si2 )(ch2−2ch sh+sh2 ) −7−12cosi−sh −ch−40 si2 −2si sh−sh2 (co+ch)2 −21 si4 −4si3sh −6si2sh2+4si sh3+sh4 (co+ch)4 ) .
The function µ(S) is plotted in figure 5 while the large N expectation EN,bS N1/4c L/N
1/2=
µ(S)/ρ(S) is represented in figure 6. At small S, we have the behavior
µ(S) = 171 10 r 3 πΓ 9 4 S4+ O(S5) so that (28) EN,bS N1/4c L/N1/2 ∼ N →∞ 399 160 r 3 πΓ 9 4 S + O(S2) .
This limiting behavior is plotted in figure 6. Returning to unscaled variables L and s, it means that we have the large N behavior
(29) EN,s(L) ∼ N →∞ 399 160 r 3 πΓ 9 4 s× N1/4 for 1 s N1/4 .
lim
N→∞1
S
E
N,b
S N1/4c
1
− e
−λ L/N 1/2!λ
S
ψ(λ)
Figure 7. A plot of the large N limit of S1 EN,bS N1/4c
1− e−λ L/N1/2 , which emphasizes the small S linear behavior (30).
More generally, we find by expandingF(S, a, λ) at small S the linear behavior:
(30) EN,bS N1/4c 1− e−λ L/N1/2 ∼ N →∞ψ(λ) S + O(S 2) ψ(λ) = 133 400 r 3 2π Z ∞ 0 dt t e−t2λ3/2 15t 2+4λ2 sr 1+4t 2 λ2−1−4t λ sr 1+4t 2 λ2+1 .
This behavior is emphasized in figure 7 upon plotting lim
N →∞ 1 S EN,bS N1/4c 1− e−λ L/N1/2 as a function of S and λ.
As for the large S behavior of ρ(S) and µ(S), a simple saddle point estimate of the corresponding contour integrals (in the a variable) shows that
ρ(S)S→∞∼ 48× 21/6 35/6S5/3e−3(32) 2/3 S4/3 , µ(S) ∼ S→∞22× √ 6 S e−3(32) 2/3 S4/3 ,
so that we get the large S behavior (31) EN,bS N1/4c L/N1/2 ∼ N →∞ 11 12× 22/331/3 1 S2/3 + O 1 S2 . This limiting behavior is plotted in figure 6.
5. Infinite maps: from scaling functions to the local limit
5.1. Extracting the local limit from scaling functions. The local limit corresponds again to bi-pointed planar quadrangulations with a fixed volume N and a fixed distance 2s between the two marked points, but in a regime where we now take the limit N→ ∞ while keeping s finite. It is now a general statement that we may describe properties of the local limit when s, although remaining finite, becomes large in terms of the scaling functions that we already computed. Let us discuss this statement in more details.
The generating function Fs(g, z) is singular when g tends to its critical value 1/12. For
g→ 1/12−, we have an expansion of the form6
(32) Fs(g, z) =
X
i≥0
fi(s, z) (1− 12 g)i/2
involving half-integer powers of (1− 12 g), and with f1(s, z) = 0.
Using the identification (1− 12 g)1/2 = (a/√6)22 whenever g = G(a, ), we may then
connect the scaling functionF(S, a, λ) to the above expansion and write F(S, a, λ) = lim→0 13FbS/c(G(a, ), Z(λ, )) = lim →0 X i≥0 a √ 6 2i 2i−3f i(bS/c , Z(λ, ))
with G(a, ) and Z(λ, ) as in (7). Now Z(λ, ) depends only on the product λ 2 so that
fi(bS/c , Z(λ, )) depends on only two variables S/ and λ 2, or equivalently on S/ and
(S/)2
× λ 2= λ S2. The existence of a non-trivial finite limit when
→ 0 implies that (33) fi(bS/c , Z(λ, )) ∼ →0 S 2i−3 ϕi(λ S2)
for i6= 1. Plugging back this expression in F(S, a, λ), we then have the direct identification F(S, a, λ) =X i≥0 a √ 6 2i S2i−3ϕ i(λ S2) ⇔ F S, a, τ S2 =X i≥0 a √ 6 2i S2i−3ϕ i(τ )
with ϕ1 = 0. We may therefore read the functions ϕi(τ ) from the expansion in S of
F(S, a, τ/S2) via (34) ϕi(τ ) = a √ 6 −2i [S2i−3] FS, a, τ S2 = [S2i−3] FS,√6, τ S2 , where we eventually set a =√6 since ϕi(τ ) does not depend on a.
Returning to the local limit, we expect that, for N→ ∞ and for finite s, the Vorono¨ı cell perimeter L will remain finite. Moreover, as we shall see, L scales as s2 at large s in the
sense that the limiting expected value Φs(ω)≡ lim
N →∞EN,s
e−ωL/s2
has a non-trivial limit when s→ ∞. Let us indeed see how to estimate Φs(ω) at large s
from the above correspondence. From (22), we have
Φs(ω) = lim N →∞ [gN]F s g, e−ω/s2 [gN]F s(g, 1)
with a leading singularity of Fs(g, z) when g → 1/12− corresponding to the i = 3 term
in (32) since the i = 0 and i = 2 terms are regular and the i = 1 term is missing. This immediately yields the large N estimate
[gN]F s(g, z) ∼ N →∞ 3 4 12N √πN5/2f3(s, z)
6It is easily checked by expanding (2) in powers of (1 − 12 g)1/4that X
s,t(g, z) has an expansion which
involves only even powers of (1−12 g)1/4, i.e. half-integer powers of (1−12 g), with moreover no term of order
(1 − 12 g)1/2. This property implies immediately a similar property for Fs(g, z) = ∆s∆tlog(Xs,t(g, z))|t=s.
The fact that f1(s, z) = 0 may also be understood by noting that f1(s, 1) = 0, as easily checked from the
explicit expression (5), and that 0 ≤ [gN]F
s(g, z) ≤ [gN]Fs(g, 1) for 0 ≤ z ≤ 1. Having a non-vanishing value
of the leading singularity coefficient f1(s, z) when 0 ≤ z < 1 would indeed contradict this latter bound as it
would imply that [gN]F
from which we deduce the identity Φs(ω) = f3 s, e−ω/s2 f3(s, 1) .
Its large s behavior may be obtained as follows: setting λ = ω/S2and = S/s in (33), with
→ 0 when S is fixed and s → ∞, we have, for i = 3, f3 s, Z(ω/S2, S/s) ∼ s→∞s 3 ϕ3(ω) = s3[S3]F S,√6, ω S2 from (34). At large s, Z(ω/S2, S/s) ∼ e−ω/s2
and we arrive immediately at the desired asymptotic expression (35) Φs(ω) ∼ s→∞ [S3] FS,√6, ω S2 [S3]F(S,√6, 0)
in terms of the scaling functionF(S, a, λ) only.
5.2. Explicit expressions and plots. From the explicit expression (20)–(21), it is a rather straightforward exercise to expand F S,√6, ω/S2 in powers of S and to extract its S3
coefficient. The only non-trivial part of the exercise comes from the hypergeometric function which enters the formula (20) for eH(σ, A). Its expansion is discussed in details in Appendix A and gives rise to terms involving the exponential integral function E1(r) defined by
E1(r)≡ Z ∞ r dt t e −t .
The net result of the small S expansion7is the following fully explicit expression:
[S3] FS,√6, ω S2 = 1 9800 (Π(q))3 Λ(q) Π(q) + 6 (6q) 3Ω(q) e6qE 1(6q) , q = r ω 3 , Π(q) = 5 + 15q + 18q2+ 9q3 , Ω(q) = 1925 + 5775q + 6930q2+ 3465q3+ 2700q6+ 2754q7+ 972q8 , Λ(q) = 14000000 + 110394375q + 394807500q2+ 841185000q3+ 1180872000q4 + 1137241620q5+ 754502040q6+ 334611972q7+ 89532864q8 + 3093876q9− 17740944q10 − 17688456q11 − 8817984q12 − 1889568q13 . For ω→ 0, this expression tends to 16/7 so that we obtain eventually the desired result
(36) Φs(ω) ∼ s→∞ 1 22400 (Π(q))3 Λ(q) Π(q)+ 6 (6q) 3 Ω(q) e6qE1(6q) , q = r ω 3 . This function is plotted in figure 8 for illustration. At large ω, it has the expansion (37) lim s→∞Φs(ω) = 1 4 √ 3 1 ω3/2 − 15 16 √ 3 1 ω5/2 + 39 16 1 ω3 + O 1 ω7/2 . At small ω, its expansion reads
(38) lim s→∞Φs(ω) = 1− 4389√3 3200 √ ω +1713 560 ω + O ω3/2log(ω) .
7As expected, the small S expansion of F
S,√6, ω/S2
involves only odd powers of S, starts with a term of order 1/S3, has no term of order 1/S so that the next term is of order S, followed by the desired
lim
s→∞
Φ
s(ω)
ω
Figure 8. A plot of the function Φs(ω) at large s, as given by (36).
lim
s→∞
P
s(`)
`
Figure 9. A plot of the function probability density Ps(`) at large s, as
obtained by a numerical inverse Laplace transform of Φs(ω). Its large ` and
small ` behaviors are given by (39) and (40) respectively.
In particular, due to the √ω singular term, the expected value of L/s2 in the local limit,
given by lim N →∞EN,s L/s 2= −dωd Φs(ω) ω=0
is infinite for large s. This is consistent with the estimate (29) of the scaling limit which states that EN,s L/s2 ∼ N →∞ 399 160 r 3 πΓ 9 4 ×N 1/4 s for 1 s N 1/4
and therefore implies that EN,s L/s2
diverges as N1/4 when N
→ ∞ at fixed finite (al-though large) s.
The inverse Laplace transform (with respect to ω) of Φs(ω) is nothing but the probability
density Ps(`) for the quantity
`≡ L s2 .
It has a non-trivial limit at large s, which may be evaluated numerically from the explicit expression (36) via some appropriate inverse Laplace transform numerical tool [1, 2, 3]. The
resulting plot is shown in figure 9. For large `, we read from the small ω behavior (38) that (39) lim s→∞Ps(`)`→∞∼ 4389 6400 r 3 π 1 `3/2 .
It particular, this behavior implies that all the positive moments of this law are infinite. For small `, we read from the large ω behavior (37) that
(40) lim s→∞Ps(`)`→0∼ 1 2 r 3 π √ ` . 6. conclusion
In this paper, we presented a number of exact laws for the Vorono¨ı cell perimeter in large bi-pointed quadrangulations, both in the scaling and in the local limit. Even though the explicit expressions of these laws, when available, are quite involved, these formulas are expected to be universal and characteristic of the Brownian map (scaling limit) or the Brownian plane (local limit at large distance). In other words, the very same expressions would be obtained in the same regimes, up to some possible global rescalings (depending on the precise definition of distances, lengths and volumes), for all families of map in the universality class of the so-called pure gravity, including in particular maps with faces of arbitrary but bounded degrees.
In the local limit N → ∞ and s finite, it was shown in [7] that exactly one of the two Vorono¨ı cells becomes infinite while the other remains finite with a volume V of order s4.
The Vorono¨ı cell perimeter L, which scales as s2, is thus the length of the frontier between a
finite and an infinite cell. It is interesting to note that we found here that all the moments of the law for L/s2are infinite, in the same way as it was found in [7] that all the moments
of the law for V /s4 diverge.
Note finally that, if we do not fix the distance between the two marked vertices, we expect at large N that the typical distance will be of order N1/4and we can therefore use the results
of the scaling limit to state that the expected value of the Vorono¨ı cell perimeter in large bi-pointed quadrangulations having their two marked vertices at arbitrary even distance from each other will be asymptotically equal to γN1/2 with
γ = Z ∞ 0 dS µ(S) = 12 √π 35 . Acknowledgements
The author thanks Michel Bauer for enlightening discussions on various technical points in the computations. The author acknowledges the support of the grant ANR-14-CE25-0014 (ANR GRAAL).
Appendix A. Expansion of the hypergeometric function 2F1(1, 2A + 2, 2A + 3, σ)
We are interested in computing the small S expansion ofF(S,√6, ω/S2) up to order S3.
To this end, we need from (20) and (21) the small S expansion of the quantity Ξ(S, ω)≡ 2F1(1, 2A + 2, 2A + 3, σ) with σ = e− √ 6S and A = r 1 + ω 2S2 .
A first naive approach would consist in first expanding the term of order σn in
2F1(1, 2A +
2, 2A + 3, σ), as read in (18), for A =p1 + ω/(2S2), namely
2A + 2 n + 2A + 2 = 1− n √ 2ωS + n(2 + n) 2ω S 2+ · · · and in summing over n to get the small S expansion
2F1(1, 2A + 2, 2A + 3, σ) = 1 1− σ − 1 √ 2ω σ (1− σ)2S + 1 2ω 3σ− σ2 (1− σ)3S 2+ · · · .
Setting then, in a second step, σ = e−
√
6S and expanding at small S, we deduce
(41) Ξ(S, ω) = 1 √ 6S +· · · −√1 2ω 1 6S +· · · + 1 2ω 1 3√6S +· · · +· · ·
where it is apparent that all the terms of the expansion contribute in fact to the same leading order 1/S. This naive approach therefore fails and a proper resummation is required. To perform this resummation and properly get the desired small S expansion, we may instead rely on the differential equation satisfied by 2F1(1, 2A + 2, 2A + 3, σ), namely
σ d
dσ 2F1(1, 2A + 2, 2A + 3, σ) + (2A + 2)2F1(1, 2A + 2, 2A + 3, σ)− 2A + 2
1− σ = 0 , which immediately translates into the following equation for Ξ(S, ω):
(42) ∂ ∂SΞ(S, ω)+2 ω S ∂ ∂ωΞ(S, ω)−2 √ 6 r 1 + ω 2S2+ 1 Ξ(S, ω)+2√6 r 1 + ω 2S2 + 1 1− e−√6S = 0 .
From the previous discussion, Ξ(S, ω) is expected to have a small S leading term of order 1/S, in which case all the different terms in the differential equation above are precisely of the same order 1/S2. Writing therefore
Ξ(S, ω) = ξ−1
2√3ω 1
S/√6+ O(1) ,
where the multiplicative factors were introduced for convenience, and expanding (42) to leading order 1/S2 at small S yields the following differential equation for ξ
−1(r):
6r d
drξ−1(r)− 6(1 + r) ξ−1(r) + r = 0 , easily solved into
ξ−1(r) = k r er+1
6r e
rE 1(r) ,
where k is some arbitrary constant and E1(r) the exponential integral function
E1(r)≡ Z ∞ r dt t e −t .
When ω→ ∞, we have clearly Ξ(S, ω) → 1/(1 − e−√6 S) = 1/(√6S) + O(1) so that ξ −1(r)
must have a finite large r limit equal to 1/6. The existence of this finite limit at large r fixes the integration constant k to k = 0, so that
ξ−1(r) =
1 6r e
rE 1(r) .
Note that, from the “naive” analysis, we expect moreover the following large r expansion, read from (41): ξ−1(r) = 1 6 1−1r + 2 r2+ O 1 r3 .
These expansion is indeed recovered from the general property
E1(r) = e−r r P X p=0 p! (−r)p + O e−r rP .
To get the desired expansion ofF(S,√6, ω/S2) up to order S3, it is easily checked from (20)
and (21) that the small S expansion of Ξ(S, ω) must be push up to order S5. We therefore
write Ξ(S, ω) = 5 X i=−1 ξi 2√3ω(S/√6)i+ O(S6) .
Expanding (42) to increasing orders in S gives rise to a sequence of differential equations for the successive functions ξi(r), i = 0, 1, . . . , 5 which are easily solved, one after the other,
leading to ξ0(r) = 1 2 4 (1 + r) erE1(r)− 3 , ξ1(r) = 1 2 r 24 1 + 3r + r2erE1(r)− (47 + 23r) , ξ2(r) = 6 r2 8 r 6 + 6r + r2erE1(r)− (17 + 40r + 8r2) , ξ3(r) = 3 10 r3 480 −3 + 3r + 21r2 + 10r3+ r4erE1(r) − (−2034 + 6246r + 4323r2+ 481r3), ξ4(r) = 54 5 r4 32 r −30 + 30r + 55r2+ 15r3+ r4erE 1(r) − (−518 − 36r + 1343r2+ 448r3+ 32r4), ξ5(r) = 9 35 r5 2688(45− 45r − 135r2+ 150r3+ 120r4+ 21r5+ r6) erE 1(r) − (154608 − 393240r + 177600r2 + 271468r3+ 53755r4+ 2687r5). This provides the desired small S expansion of Ξ(S, ω) which, inserted in (20)–(21), yields the wanted expansion ofF(S,√6, ω/S2) up to order S3.
References
[1] J. Abate and P. P. Valk´o. Mathematica packages “Numerical Laplace Inversion” and “Numeri-cal Inversion of Laplace Transform with Multiple Precision Using the Complex Domain”, avail-able on the Wolfram Library Archive: http://library.wolfram.com/infocenter/MathSource/4738/ and http://library.wolfram.com/infocenter/MathSource/5026/ .
[2] J. Abate and P. P. Valk´o. Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Computers & Mathematics with Applications, 48(3):629 – 636, 2004.
[3] J. Abate and P. P. Valk´o. Multi-precision Laplace transform inversion. International Journal for Numer-ical Methods in Engineering, 60(5):979–993, 2004.
[4] J. Bouttier, P. Di Francesco, and E. Guitter. Geodesic distance in planar graphs. Nucl. Phys. B, 663(3):535–567, 2003.
[5] G. Chapuy. On tessellations of random maps and the tg-recurrence. S´eminaire Lotharingien de
Combina-toire, 78B:Article #79, 2017. Proceedings of the 29th Conference on Formal Power Series and Algebraic Combinatorics (London).
[6] E. Guitter. On a conjecture by Chapuy about Voronoi cells in large maps. J. Stat. Mech., 2017(10):103401, 2017.
[7] E. Guitter. A universal law for Voronoi cell volumes in infinitely large maps, 2017. arXiv:1706.08809. [8] G. Miermont. Tessellations of random maps of arbitrary genus. Ann. Sci. ´Ec. Norm. Sup´er. (4),
42(5):725–781, 2009.
Institut de physique th´eorique, Universit´e Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette E-mail address: emmanuel.guitter@ipht.fr
